Zero Mach Number Limit of the Compressible Primitive Equations Part I: Well-prepared Initial Data

This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order $\mathcal O(\varepsilon)$, as $\varepsilon \rightarrow 0^+$, where $\varepsilon$ represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows.